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Arithmétique des surfaces cubiques diagonales. (Arithmetic of diagonal cubic surfaces). (French) Zbl 0639.14018

Diophantine approximation and transcendence theory, Semin., Bonn/FRG 1985, Lect. Notes Math. 1290, 1-108 (1987).
Using the Brauer groups of schemes Manin has defined an obstruction to the validity of the Hasse principle for cubic surfaces [and varieties, see Yu. I. Manin, “Cubic forms. Algebra, geometry, arithmetic” (1974); translation from the Russian original edition (1972; Zbl 0255.14002); see also the 2nd enlarged edition (1986)]. The authors develop an algorithm for the computation of that obstruction in the case of diagonal surfaces ax \(3+by\) \(3+cz\) \(3+dt\) \(3=0\). They discover the following infinite series of counterexamples to the Hasse principle: \(a=1\), \(b=p\) 2, \(c=pq\), \(d=q\) 2 where \(p\equiv 2 \bmod 9\) and \(q\equiv 5 \bmod 9\) are primes. It is computed also the obstruction for all equations with \(0<a,b,c,d<100\). That gives 245 counterexamples to the Hasse principle. The known before Bremner example with \(a=1\), \(b=4\), \(c=10\), \(d=25\) belongs to the list and minimizes the product abcd.
The computations confirm the following conjecture: for rational surfaces over number fields Manin’s obstruction is the only obstruction to the validity of the Hasse principle. - In the special case of diagonal cubic surfaces V the conjecture can be given more precisely: if there exists a prime p dividing one and only one of the coefficients a,b,c,d (supposed to be integer and without cubic factor) then \(\prod_{\ell}V({\mathbb{Q}}_{\ell})\neq \emptyset\) implies that V(\({\mathbb{Q}})\neq \emptyset\).
Reviewer: A.Parshin

MSC:

14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties
14G25 Global ground fields in algebraic geometry
11D25 Cubic and quartic Diophantine equations